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This question is very closely related to my other question here.

Let $\Gamma$ be a countable discrete group and $\pi:\Gamma \rightarrow \mathcal{H}$ be a unitary representation of $\Gamma$. A map $b:\Gamma \rightarrow \mathcal{H}$ is an additive cocycle if $$b(gh)=\pi(g)b(h)+b(g)$$ for all $g,h\in G$.

If we consider the collection of all $(b,\pi)$ where $\pi$ is a unitary representation of $\Gamma$ and $b$ is a cocycle as above, can this collection "see" if $\Gamma$ is a free group?

Again, I'm being deliberately vague here regarding the word "collection". Roughly, what I am asking is whether including these cocycles in my previous question would yield a structure that can detect freeness of the underlying group.

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up vote 3 down vote accepted

I think your question is not precise enough. If you just look at this as a set or topological space, it is most likely not enough. If you add additional structure like sum and tensor product, then it is enough.

Maybe this is what you are looking for: If $\Gamma$ is free on the set $X$, then for any unitary representation $\pi \colon \Gamma \to U(H)$ any map $c \colon X \to H$ extends uniquely to a $1$-cocycle on $\Gamma$. Moreover, this happens if and only if $\Gamma$ is free. In other words, the linear space of 1-cocycles for $\pi$ on $\Gamma$ is precisely the space of $H$-valued functions on $X$.

This observation can be used to show that the first $\ell^2$-Betti number of $\Gamma$ is $|X|-1$. (Minus one, because you have to substract the inner 1-cocycles.) The first $\ell^2$-Betti number of $\Gamma$ precisely counts how many 1-cocycles with values in the left-regular representation $\lambda$ exist.

One does not even have to know this for all unitary representations $\pi$, it is enough to know it for the left-regular representation $\lambda$. Indeed, a group on $n$ generators is free if and only if its first $\ell^2$-Betti number equals $n-1$. This is sharp, as their are $n$-generated groups which are not free but the first $\ell^2$-Betti number exceed $n-1 - \varepsilon$, for example $\Gamma = \langle a,b \mid a^k \rangle$ for $k$ high enough. More interestingly, Denis Osin has constructed $n$-generated torsion groups with first $\ell^2$-Betti number higher than $n-1 - \varepsilon$.

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This is great, Andreas! (I had a feeling you'd be able to mop this up quickly!) – Jon Bannon Nov 29 '10 at 15:08

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